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Conservation of Energy
What is energy? We all know what energy is--it helps us do things, it gets things moving. It makes cars move along a highway. It makes a rollercoaster go uphill, downhill, around a curve, upside down, etc. In physics, energy is defined as an object's ability to do work, or to apply a force through a distance. Types of energy Energy can exist as potential energy or as kinetic energy. It may be converted from potential energy to kinetic energy or vice versa but it is neither created nor destroyed. a. Potential Energy The gravitational potential energy of an object is the work done by a constant gravitational force F = mg (weight) on the object to move it from one position to another position. Potential energy is defined as PE = mgh. m represents the mass of the object, g represents the acceleration of gravity (approximately 9.8 m/s/s on Earth) and h represents the height of the object. Or, mg represents Fg, the force needed to overcome gravity, or more simply, the weight of the object and h is the vertical displacement. Potential energy depends on height, so the higher up an object is, the greater its potential energy. More on Potential Energy b. Kinetic Energy An object's kinetic energy is the energy that a body possesses due to its motion. Kinetic energy is defined as . Since kinetic energy is energy from motion, it depends on velocity, so the faster an object is, the greater its kinetic energy. More on Kinetic Energy c. Interaction of Gravitational Potential Energy and Kinetic Energy Potential energy gets converted into kinetic energy, and likewise, kinetic energy gets converted into potential energy. Say you are skiing uphill. Ignoring air resistance, you are doing work, by going uphill--this work is really your kinetic energy. As you go higher and higher up the mountain, your potential energy is increasing since your height on the mountain is increasing. Also while you go higher and higher up, your speed decreases to zero which means that your kinetic energy is decreasing since you slow down. What's really happening is that the potential energy is transforming into kinetic energy. On your trip down from the top of the mountain, you go more and more downhill so your height decreases. That means that your potential energy decreases as well. Since the potential energy decreases, the kinetic energy increases. Because the total energy in the system must stay the same, a change in potential energy results in an opposite change in kinetic energy (and vice versa) to "undo" or compensate for that change, thus maintaining the total energy of the system. d. Mechanical Energy Mechanical Energy is the energy that an object has due to its stored energy (potential energy) or due to its motion (kinetic energy). Objects possessing mechanical energy can do work. The total mechanical energy of a closed system, Et = PE + KE. The Law of Conservation of Energy states that an object's mechanical energy remains constant in a closed system. More on Mechanical Energy What is the Conservation of Energy Law? Imagine that you have an empty jar. Now, imagine filling it up with jellybeans and then tightly sealing it so that the jar remains shut no matter what. You can shake the jar, roll it around, toss it up in the air, run around in a circle with it, etc., but since the jar is sealed tightly shut the jellybeans will remain safe and secure inside the jar. No jellybeans will escape and no more jellybeans will be added inside the jar. So, the amount of jellybeans you originally put inside the jar is the amount of jellybeans you end up with. Essentially, you don't lose any jellybeans and you don't gain any jellybeans--the amount of jellybeans remains the same. Similarly, this concept exists in physics as the conservation of energy. The Conservation of Energy Law states that energy is neither created nor destroyed. In an ideal closed system, where no external factors (i.e...friction) act, the total mechanical energy remains constant. There are two important things to remember: 1. To reiterate, the energy of a closed system remains the same in the beginning and in the end. Initial Potential Energy + Initial Kinetic Energy = Final Potential Energy + Final Kinetic Energy, or more simply stated PEi + KEi = PEf + KEf. 2. The total energy of a system is the sum of its potential energy and its kinetic energy. Total Energy = Potential Energy + Kinetic Energy or Et = PE + KE. Historical Development of the Conservation of Energy Law Of course, many early pooping philosophers and scientists had begun developing the Conservation of Energy Law but a German man named Gottfried Wilhelm Leibniz first attempted to express this principle using a mathematical equation between 1676 and 1689. Leibniz observed that in many mechanical systems with several masses mi, each with velocity vi, stayed the same. He called this value the vis viva or living force of the system. This accurately represents the approximate conservation of kinetic energy in most situations. Many physicists however pretty much ignored Leibniz's principle since renwoned physicists Sir Issac Newton and René Descartes had established the conservation of momentum principle: Meanwhile, engineers, including John Smeaton, Peter Ewart, Gustave-Adolphe Hirn and Marc Seguin used Leibniz's principle, arguing that conservation of momentum by itself was insufficient in practical calculations. Scientists came to realize that kinetic energy is not always conserved; sometimes energy gets "lost". Gradually, they came to an agreement that heat inevitably created by motion was another form of the living force of the system, vis viva. In 1798, scientist Benjamin Thompson, better known as Count Rumford, observed that heat generated during the boring of cannons. This offered more proof that physical motion could be converted into heat. In 1807, English physician and physicist, Thomas Young became the first man to use and define the term energy as it is now known in physics. Thereafter, Vis viva was referred to as energy. From 1819-1839, French scientist Gaspard-Gustave Coriolis and French mathematician Jean-Victor Poncelet and revised the equation of Vis viva to a more accurate equation In 1837, German pharmacist Karl Friedrich Mohr published a paper, Über die Natur der Wärme, in the Zeitschrift für Physik. He wrote, "besides the 54 known chemical elements there is in the physical world one agent only, and this is called Kraft energy. It may appear, according to circumstances, as motion, chemical affinity, cohesion, electricity, light and magnetism; and from any one of these forms it can be transformed into any of the others." This is a simple statement of an important component of the Conservation of Energy Law, that energy can be converted from one form to another. In 1840, German surgeon Julius Robert von Mayer found out that heat and mechanical work were both forms of energy. He was on a voyage from the East Indies back to Germany and was treating sick sailors on the boat. He noticed that a lot of sailors had unusually dark red blood. Then he realized that in the tropics, since it's hotter there, people get less oxygen, and therefore less energy, to maintain their body temperatures. Also, people burn less of the food (the caloric energy) that they eat and therefore, they don't generate as much heat or energy. He knew that he realized something very important, that heat and work were interchangeable and could be traded back and forth. But he wasn't a physicist so he spent time studying physics before he was able to come up with a quantitative relationship between heat and work. The scientific community rejected his work. Meanwhile, in 1843 James Prescott Joule conducted several experiments to study heat. Through these experiments, he discovered the relationship of heat and work. In his most famous experiment, called the "Joule apparatus", he had a falling weight attached to a string to spin a paddle that was inside an insulated tub of water. He measured the increasing water temperature and showed that the gravitational potential energy that the weight lost everytime it fell more into the water was the same as the thermal energy (heat) gained by the water from friction generated with the paddle. Although the scientific community did not really accept his work, he got more recognition than Mayer did. In 1847, German physicist Hermann von Helmholtz published a book Über die Erhaltung der Kraft (On the Conservation of Force), where he established a relationship between mechanics, heat, light, electricity and magnetism by saying that all of them were a single force--energy. He relied on earlier works by others such as Mayer and Joule. He established the Conservation of Energy law, and the scientific community generally embraced it. A Little Bit on Thermodynamics Thermodynamics is a branch of physics that investigates the movement of energy and how energy causes movement in the universe. It studies the relationships that exist among heat, work and energy and the conversions among them in a large system, the universe. Thermodynamics is based on observations of nature and experiences in nature. It is very practical and important because many machines and modern devices change heat into work (such as an automobile engine) or turn work into heat (or cooling, as in a refrigerator). You can think of thermodynamics as a fancy way to say the conversion of energy. Essential to thermodynamics is the Conservation of Energy principle. The First Law of Thermodynamics is basically the same thing as the Conservation of Energy Law. It states that the change in a system's internal energy equals the difference of the work the system does (or is has done on it) and the heat that the system absorbs (or releases). It states that energy cannot be created or destroyed. Instead it is converted from one form to another, such as from work to heat, from heat to light, from chemical to heat, or such. One example is how the kinetic energy of a moving car is converted into heat energy at the brakes and tire surfaces. Or, for an example that you can relate to your own home, think of how a toaster converts electric energy into heat to cook bread for your morning toast. (Note: Enthalpy is the sum of the internal energy of matter and the product of its volume and pressure.) For more information on the derivation of the first law of thermodynamics, feel free to visit Thermodynamics Check your understanding! Here are some review questions relating to the Conservation of Energy for you to check your understanding: a. Consider the falling motion of the ball in the following two frictionless situations. For each situation, indicate what type of forces are doing work upon the ball. Indicate whether the energy of the ball is conserved and explain why. Finally, indicate the kinetic energy and the velocity of the 2-kg ball just prior to striking the ground. For questions b and c, refer to the image below: b. As the object moves from point A to point D across the frictionless surface, what happens to the sum of its gravitational potential and kinetic energies? c. When will the object have minimal potential energy? Answers: a. Gravity is the only force that is doing work. It is acting on the system internally, so the total mechanical energy remains conserved. Therefore, all 100J of Potential Energy is converted to Kinetic Energy when the ball hits the ground, at the bottom of the hill. Using the equation , we can determine that the velocity is 10 m/s^2. b. The sum of the object's potential and kinetic energies (it's mechanical energy) remains the same throughout its path of travel whenever no external forces (such as friction) are acting on it. c. At Point B, the object will have minimal potential energy. Potential energy depends on height so when the height is lowest, the potential energy is lowest as well. When the potential energy is low, the kinetic energy is high. References The following websites were referred to throughout this page: 1. http://en.wikipedia.org/wiki/Potential_energy 2. http://en.wikipedia.org/wiki/Kinetic_energy 3. http://www.glenbrook.k12.il.us/gbssci/phys/class/energy/u5l1d.html 4. http://www.egs.edu/resources/leibniz.html 5. http://www.newton.cam.ac.uk/newtlife.html 6. http://www.maths.tcd.ie/pub/HistMath/People/Descartes/RouseBall/RB_Descartes.html 7. http://www.asce.org/history/bio_smeaton.html 8. http://en.wikipedia.org/wiki/Peter_Ewart 9. http://www.answers.com/topic/gustave-adolphe-hirn 10. http://en.wikipedia.org/wiki/Marc_S%C3%A9guin 11. http://www.rumford.com/Rumford2.html 12. http://www.whonamedit.com/doctor.cfm/1715.html 13. http://en.wikipedia.org/wiki/Gaspard-Gustave_Coriolis 14. http://en.wikipedia.org/wiki/Jean-Victor_Poncelet 15. http://en.wikipedia.org/wiki/Karl_Friedrich_Mohr 16. http://www.uh.edu/engines/epi722.htm 17. http://en.wikipedia.org/wiki/James_Prescott_Joule 18. http://en.wikipedia.org/wiki/Hermann_von_Helmholtz 19. http://www.grc.nasa.gov/WWW/K-12/airplane/thermo1f.html Resources For more information, you can go to these websites: 1. http://www.glenbrook.k12.il.us/gbssci/phys/Class/energy/u5l1b.html 2. http://www.glenbrook.k12.il.us/gbssci/phys/Class/energy/u5l1c.html 3. http://www.grc.nasa.gov/WWW/K-12/airplane/thermo1f.html 4. http://www.school-for-champions.com/science/thermodyn.htm 5. Barron's Regents Review for Physics, by Miriam A. Lazar, third Edition. Barron's Educational Series, Inc. 2004 (This book is great for a general overview of physics concepts, especially those found on the NYS Regents exam)